In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal.
I would like a proof or a significant hint to establish that on the sphere, this is 1 plus (or minus in odd dimensions) the degree (winding number) of that map.
This looks like essentially the same question as Exercise concerning the Lefschetz fixed point number for the special case of a sphere.